Locally analytic vectors of unitary principal series of <span class="mathjax-formula">${\mathrm {GL}}_2({\mathbb {Q}}_p)$</span>

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(1)ISSN 0012-9593. ASENAH. quatrième série - tome 45. fascicule 1. janvier-février 2012. aNNALES SCIENnIFIQUES de L ÉCOLE hORMALE SUPÉRIEUkE. Ruochuan LIU & Bingyong XIE & Yuancao ZHANG. Locally analytic vectors of unitary principal series of GL2(Qp). SOCIÉTÉ MATHÉMATIQUE DE FRANCE.

(2) Ann. Scient. Éc. Norm. Sup.. 4 e série, t. 45, 2012, p. 167 à 190. LOCALLY ANALYTIC VECTORS OF UNITARY PRINCIPAL SERIES OF GL2 (Qp ).  R LIU, B XIE  Y ZHANG. A. – The p-adic local Langlands correspondence for GL2 (Qp ) attaches to any 2-dimensional irreducible p-adic representation V of GQp an admissible unitary representation Π(V ) of GL2 (Qp ). The unitary principal series of GL2 (Qp ) are those Π(V ) corresponding to trianguline representations. In this article, for p > 2, using the machinery of Colmez, we determine the space of locally analytic vectors Π(V )an for all non-exceptional unitary principal series Π(V ) of GL2 (Qp ) by proving a conjecture of Emerton. R. – La correspondance de Langlands locale p-adique pour GL2 (Qp ) associe à toute représentation irréductible p-adique V de dimension 2 de GQp une représentation admissible unitaire Π(V ) de GL2 (Qp ). Les séries principales unitaires de GL2 (Qp ) sont les Π(V ) correspondant aux représentations triangulines. Dans le présent article, en utilisant la machinerie de Colmez, on détermine l’espace des vecteurs localement analytiques Π(V )an pour toute série principale unitaire non-exceptionnelle Π(V ) de GL2 (Qp ), et on démontre ainsi une conjecture d’Emerton.. 1. Introduction Let F be a finite extension of Qp . The aim of the p-adic local Langlands programme initiated by Breuil is to look for a “natural” correspondence between certain n-dimensional p-adic representations of Gal(Qp /F ) and certain Banach space representations of GLn (F ). Thanks to much recent work, especially that of Colmez and Paškūnas, we have gained a fairly clear picture in the case F = Qp and n = 2 which is the so-called p-adic local Langlands correspondence for GL2 (Qp ) establishing a functorial bijection between 2-dimensional irreducible p-adic representations of Gal(Qp /Qp ) and non-ordinary irreducible admissible unitary representations of GL2 (Qp ). Although the present version of p-adic local Langlands correspondence is formulated at the level of Banach space representations, it is very useful, as in Breuil’s initial work [4], to have information of the subspace of locally analytic vectors. Fix a finite extension L of Qp as the coefficient field, and we denote by Π(V ) the corresponding unitary representation of GL2 (Qp ) for any 2-dimensional irreducible L-linear representation V of Gal(Qp /Qp ). 0012-9593/01/© 2012 Société Mathématique de France. Tous droits réservés ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE.

(3) 168. R. LIU, B. XIE AND Y. ZHANG. The unitary principal series of GL2 (Qp ), which are the simplest ones among these Π(V ), are those corresponding to trianguline representations. In [13], Emerton made a conjectural description of the subspace of locally analytic vectors Π(V )an for all unitary principal series Π(V ). We recall his conjecture below. Let Sirr be the parameterizing space of 2-dimensional irreducible trianguline representations of Gal(Qp /Qp ) introduced by Colmez in [7]. For any s ∈ Sirr , let V (s) be the corresponding trianguline representation. We may write s = (δ1 , δ2 , L ) so that the étale (ϕ, Γ)-module Drig (V (s)) is isomorphic to the extension of R(δ2 ) by R(δ1 ) defined by L . For any such s, if δ1 δ2−1 = xk |x| for some k ∈ Z+ , then we set Σ(s) to be the locally analytic 2−k GL2 (Qp )-representation Σ(k + 1, L ) ⊗ ((δ2 |x| 2 ) ◦ det) where {Σ(k + 1, L )} is the family of locally analytic GL2 (Qp )-representations introduced by Breuil in [4]. Otherwise, Ä GL (Q ä ) 2 we define Σ(s) to be the locally analytic principal series IndB(Qp ) p δ2 ⊗ δ1 (x|x|)−1 . The an conjecture of Emerton is: C 1.1 ([13, Conjecture 6.7.3, 6.7.7]). – For any s ∈ Sirr , Π(V (s))an sits in an exact sequence Ä GL (Q ) ä (1.1) 0 −→ Σ(s) −→ Π(V (s))an −→ IndB(Q2p ) p δ1 ⊗ δ2 (x|x|)−1 −→ 0. an. In the special cases when V (s) are twists of crystabelian representations and nonexceptional, there is a more precise conjectural description of Π(V (s))an due to Breuil. In [16], the first author proved Breuil’s conjecture. The main result of this paper is: T 1.2 (Theorem 6.13). – For p > 2, Conjecture 1.1 is true if V (s) is nonexceptional. In fact, one can easily deduce Breuil’s conjecture from Emerton’s conjecture. Thus for p > 2, the above theorem covers the aforementioned result of the first author. We now explain the strategy of the proof of Theorem 1.2. The whole proof builds on Colmez’s machinery of p-adic local Langlands correspondence for GL2 (Qp ) developed in [12]. The key ingredient is Colmez’s identification of the locally analytic vectors: \ (Π(V̌ )an )∗ = Drig P1. (1.2). where D = D(V ) is Fontaine’s étale (ϕ, Γ)-module associated to V . To apply this formula, × 1 for any continuous characters δ, η : Q× p → L , we construct the objects R(η) δ P + 1 and R (η) δ P which areÄequipped with continuous ä GL2 (Qp )-actions, and the latter is GL2 (Qp ) −1 −1 topologically isomorphic to ( IndB(Qp ) δ η ⊗ η )∗ . In doing so, we are led to modify an some of Colmez’s constructions to twists of étale (ϕ, Γ)-modules and rank 1 (ϕ, Γ)-modules. ` On the other hand, Colmez [9] (for p > 2 and s ∈ S∗ng S∗st ; this is the only place where we need p > 2) and Berger-Breuil [3] (for s ∈ Sirr non-exceptional) establish an explicit isomorphism As : Π(V (s)) ∼ = Π(s) (for s exceptional, Paškūnas proves Π(V (s)) ∼ = Π(s) by an indirect method [17]) where Π(s) is the unitary principal series associated to V (s). We deduce from the explicit description of As plus a duality argument the following exact sequence (1.3). 0 −→ R(δ1 ) P1 −→ Drig (V (s)) P1 −→ R(δ2 ) P1 −→ 0.. 4 e SÉRIE – TOME 45 – 2012 – No 1.

(4) LOCALLY ANALYTIC VECTORS. 169. Ä GL (Q ) ä Then the natural inclusion IndB(Q2p ) p δ2 ⊗ δ1 (x|x|)−1 ,→ Π(s)an induces the following an commutative diagram (Π(s)an )∗. / D\ (š) P1 rig. ä ä∗ ÄÄ GL (Q ) 2 p IndB(Qp ) δ2 ⊗ δ1 (x|x|)−1. / R(δ̌1 ) P1. (1.4). an. where š = (δ̌2 , δ̌1 , L ) corresponds to the Tate dual of V (s). Using (1.4) together with the \ \ (š) P1 are orthogonal complements of each other under the (s) P1 and Drig fact that Drig paring {·, ·}P1 : Drig (V (s)) P1 × Drig (V (š)) P1 → L, and that (1.3) is dual to (1.5). 0 −→ R(δ̌2 ) P1 −→ Drig (V (š)) P1 −→ R(δ̌1 ) P1 −→ 0,. \ we deduce that if δ1 δ2−1 6= xk |x| for any k ∈ Z+ , then Drig (š) sits in the exact sequence. (1.6). \ 0 −→ R + (δ̌2 ) P1 −→ Drig (š) P1 −→ R + (δ̌1 ) P1 −→ 0.. We therefore conclude (1.1) by taking the dual of (1.6). If δ1 δ2−1 = xk |x| for some k ∈ Z+ , \ we have that the image of Drig (š) in R(δ̌1 ) P1 is a closed subspace of R + (δ̌1 ) P1 of \ codimension k and R(δ̌2 ) P1 ∩ Drig (š) P1 contains R + (δ̌2 ) P1 as a closed subspace of codimension k. We then conclude (1.1) using Schneider and Teitelbaum’s results on the Jordan-Hölder series of locally analytic principal series of GL2 (Qp ). The organization of the paper is as follows. In §2, we recall some necessary background of the theory of (ϕ, Γ)-modules. In §3, we recall some of Colmez’s constructions of the p-adic local Langlands correspondence for GL2 (Qp ) especially his identification of the locally analytic vectors, and we make the aforementioned modification. We review the isomorphism As : Π(V (s)) ∼ = Π(s) in §4. In §5.1, we recall Schneider and Teitelbaum’s results on Jordan+ 1 Hölder series of the analytic principal Ä locally ä series of GL2 (Qp ). We prove that R (η) δ P GL2 (Qp ) −1 is isomorphic to ( IndB(Qp ) δ η ⊗ η −1 )∗ in §5.2. Section 6 is devoted to the proof of an Theorem 1.2. We first recall the definition of Σ(s) and restate Emerton’s conjecture in §6.1. Then we prove the exact sequence (1.3) in §6.2. We finish the proof of Theorem 1.2 in §6.3. After the work presented in this paper was finished, we learned from Colmez that he had a proof of Conjecture 1.1 for all p and all trianguline representations of GQp . The strategy of his Ä GL (Q ) ä proof is different from ours. He constructs the map Π(s)an → IndB(Q2p ) p δ1 ⊗ δ2 (x|x|)−1 an directly by computing the module de Jacquet dual of Π(s)an . We refer the reader to his paper [10] for more details. Notation and conventions Let p be a prime number, and let vp denote the p-adic valuation on Qp , normalized by vp (p) = 1; the corresponding norm is denoted by | · |. Let GQp denote Gal(Qp /Qp ) for simplicity. Let χ : GQp → Z× p be the p-adic cyclotomic character. The kernel of χ is H = Gal(Qp /Qp (µp∞ )), and let Γ = Gal(Qp (µp∞ )/Qp ). For any a ∈ Z× p , let σa be the ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE.

(5) 170. R. LIU, B. XIE AND Y. ZHANG. unique element in Γ such that χ(σa ) = a. For any positive integer h, let Γh = χ−1 (1+ph Zp ). If we regard χ as a character of Q× p via the local Artin map, then it is equal to (x) = x|x|. Throughout this paper, we fix a finite extension L of Qp . We denote by OL the ring of intec(L) be the set of all continuous characters gers of L, and by kL the residue field. Let T × c(L), the Hodge-Tate weight w(δ) of δ is defined δ : GQp → L . For any δ ∈ T log δ(u) by w(δ) = log u where u is any element of Q× p \ µp∞ . For any L-linear representation ∗ V of GQp , we denote by V̌ the Tate dual V () of V . For any L ∈ L, let logL : Q× p → L be +∞ P (1−x)n the homomorphism defined by logL (p)=1 and logL (x) = − when |x − 1| < 1. n n=1. We put log∞ = vp . Hence logL is defined for all L ∈ P1 (L). Let B be the subgroup of upper triangular matrices of GL2 , let P = [ ∗0 ∗1 ] be the mirabolic subgroup of GL2 , let T be the subgroup of diagonal matrices of GL2 , and let Z be the center of GL2 . Put w = [ 01 10 ]. Let Reptors GL2 (Qp ) be the category of smooth OL [GL2 (Qp )]-modules which are of finite lengths and admit central characters. Let Rep OL GL2 (Qp ) be the category of OL [GL2 (Qp )]-modules Π which are separated and complete for the p-adic topology, p-torsion free, and Π/pn Π ∈ Reptors GL2 (Qp ) for any n ∈ N. Acknowledgements The first author thanks Christophe Breuil and Pierre Colmez for helpful discussions and communications, and Liang Xiao for helpful comments on earlier drafts of this paper. The second and the third authors thank Qingchun Tian for helpful discussions and encouragements. The first author would also like to thank Henri Darmon for his various help during the first author’s stay at McGill university. The first author wrote this paper as a postdoctoral fellow of Centre de Recherches Mathématiques. Part of this work were done while the first author was a visitor at Institut des Hautes Études Scientifiques and Beijing International Center for Mathematical Research. The first author is grateful to these institutions for their hospitality. While writing this paper, the second author was supported by the postdoctoral grant 533149-087 of Peking University and Beijing International Center for Mathematical Research.. 2. Preliminaries on (ϕ, Γ)-modules 2.1. Dictionary of p-adic functional analysis P Let OE be the ring of Laurent series f = i∈Z ai T i , where ai ∈ OL , such that vp (ai ) → ∞ as i → −∞. Let E = OE [1/p] be the fraction field of OE . P i For any r ∈ R+ ∪ {+∞}, let E ]0,r] be the ring of Laurent series f = i∈Z ai T , with ai ∈ L, such that f is convergent on the annulus 0 < vp (T ) ≤ r. For any 0 < s ≤ r ≤ +∞, we define the valuation v {s} on E ]0,r] by v {s} (f ) = inf {vp (ai ) + is} if s 6= ∞; v {∞} (f ) = vp (f (0)). i∈Z. We provide E with the Fréchet topology defined by the family of valuations S {v {s} |0 < s ≤ r}; then E ]0,r] is complete. We equip the Robba ring R = r>0 E ]0,r] ]0,r]. 4 e SÉRIE – TOME 45 – 2012 – No 1.

(6) 171. LOCALLY ANALYTIC VECTORS. with the inductive limit topology. We denote E ]0,+∞] , the ring of analytic functions on the open unit disk, by R + . Let E † be the subring of overconvergent elements of E , i.e. E † is the set of f ∈ E such that f (T ) is convergent over some annulus 0 < vp (T ) ≤ r. Let E (0,r] = E † ∩ E ]0,r] . We S equip E † = r>0 E (0,r] with the inductive limit topology. We denote E (0,∞] = OL [[T ]][1/p] by E + , and let OE + = OL [[T ]]. Let R denote any of OE + , E + , OE , E , E † , R + and R. We equip the ring R with commuting continuous actions of ϕ and Γ defined by ϕ(f (T )) = f ((1 + T )p − 1),. (2.1). γ(f (T )) = f ((1 + T )χ(γ) − 1),. γ ∈ Γ.. If we view R as a ϕ(R)-module, then R is freely generated by {(1 + T )i |i = 0, . . . , p − 1}. p−1 P (1 + T )i ϕ(yi ) for some uniquely determined Thus for any y ∈ R, we may write y = i=0. y0 , . . . , yp−1 ∈ R, and we define the operator ψ : R → R by setting ψ(y) = y0 . It follows directly from the definition that ψ is a left inverse to ϕ, and that ψ commutes with P the Γ-action. For any f = i∈Z ai T i ∈ R, we define the residue of the 1-form ω = f · dT dT ). as res(ω) = a−1 ; and for any f ∈ R, we define res0 (f ) = res(f 1+T We denote by C 0 (Zp , L) the space of continuous functions on Zp with values in L; this is an L-Banach space equipped with the supremum norm. Let LA(Zp , L) denote the space of locally analytic functions on Zp with values in L. The classical results of Mahler and Amice assert that the set of functions { nx }n∈N constitutes an orthogonal basis of C 0 (Zp , L), and P that for f = n∈N an (f ) nx ∈ C 0 (Zp , L), f ∈ LA(Zp , L) if and only if there exists some r > 0 such that vp (an (f )) − rn → +∞ as n → +∞. For any u ≥ 0, we denote by C u (Zp , L) the space of all C u -functions on Zp ; this is an L-Banach space (see [8] for more details). We have LA(Zp , L) ⊂ C u (Zp , L) ⊆ C 0 (Zp , L), and that LA(Zp , L) is dense in C u (Zp , L) for any u ≥ 0. We denote by D(Zp , L), Du (Zp , L) the topological dual of LA(Zp , L), C u (Zp , L) respectively. Note that the natural map Du (Zp , L) → D(Zp , L) is injective since LA(Zp , L) is dense in C u (Zp , L). The elements of D(Zp , L) are called distributions on Zp . A distribution µ is called of order u if µ ∈ Du (Zp , L). We define the actions of ϕ, ψ and Γ on D(Zp , L) by the formulas Z Z Z Z Z Z f ϕ(µ) = f (px)µ, f ψ(µ) = f (p−1 x)µ, f σa (µ) = f (ax)µ Zp. Zp. Zp. pZp. Zp. Zp. for any f ∈ LA(Zp , L), µ ∈ D(Zp , L) and a ∈ Z× p. The Amice transformation A on D(Zp , L) is defined by Z Z +∞ X A : D(Zp , L) → L[[T ]], A (µ) = (1 + T )x µ(x) = Tn Zp. n=0. Zp. x n. ! µ.. It is an immediate consequence of the results of Mahler and Amice that the Amice transformation µ 7→ A (µ) induces topological isomorphisms from D0 (Zp , L) and D(Zp , L) to E + and R + respectively which are compatible with the actions of ϕ, ψ and Γ. We denote by LAc (Qp , L) the space of compactly supported L-valued locally analytic functions on Qp , and denote by D(Qp , L) the topological dual of LAc (Qp , L). The elements ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE.

(7) 172. R. LIU, B. XIE AND Y. ZHANG. of D(Qp , L) are called distributions on Qp . For any µ ∈ D(Qp , L), let µ(n) be the distribution on Zp defined by Z Z f µ(n) = f (pn x)µ Zp. p−n Zp. for any f ∈ LA(Zp , L). It follows that ψ(µ(n+1) ) = µ(n) , and that any sequence of distributions (µ(n) )n∈N on Zp so that ψ(µ(n+1) ) = µ(n) uniquely determines a distribution µ on Qp . The Amice transformation A (µ) for µ ∈ D(Qp , L) is then defined to be the sequence (A (µ(n) ))n∈N . A distribution µ on Qp is said to be of order u if all µ(n) are of order u. The distribution µ is said to be globally of order u, if there is a constant Cu (µ) such that vDu (µ(n) ) ≥ nu+Cu (µ) for all n ∈ N. Let Du (Qp , L) denote the space of distributions on Qp globally of order u. 2.2. The category of (ϕ, Γ)-modules Keep notations as in §2.1. We define a (ϕ, Γ)-module over R to be a finite free R-module D equipped with commuting continuous semilinear ϕ, Γ-actions. When R = OE , the (ϕ, Γ)-module D is called étale if ϕ(D) generates D as an OE -module. When R = E , the (ϕ, Γ)-module D is called étale if it arises by base change from an étale (ϕ, Γ)-module over OE . A (ϕ, Γ)-module D over E † is called étale if D ⊗E † E is étale as an (ϕ, Γ)-module over E . A (ϕ, Γ)-module D over R is called étale if the underlying ϕ-module is pure of slope 0 in the sense of Kedlaya’s slope theory [15]. c(L), we define R(δ) to be the rank 1 (ϕ, Γ)-module E 2.1. – For any δ ∈ T over R which has an R-base e satisfying (2.2). ϕ(e) = δ(p)e,. σa (e) = δ(a)e,. a ∈ Z× p.. Such an element e, which is unique up to a nonzero scalar (this is because Rϕ=1,Γ=1 = L or OL ), is called a standard basis of R(δ). Let V be a d-dimensional L-linear representation of GQp , and let T be a GQp -invariant ur be the p-adic completion of the maximal unramified extension OL -lattice of V . Let Ed ur ur . The ϕ, Γ-actions on E naturally extend to of E , and let d OE be the ring of integers of Ed ur . We define d continuous actions, which we again denote by ϕ, Γ respectively, on E H D(T ) = (T ⊗ OE d Our E ). ur )H ), d (resp. D(V ) = (V ⊗E E. which is a (ϕ, Γ)-module over OE (resp. E ). We define D† (V ) to be the maximal finite dimensional ϕ, Γ-stable E † -subspace of D(V ), and we define Drig (V ) = D† (V ) ⊗E † R; then D† (V ) and Drig (V ) are (ϕ, Γ)-modules over E † and R respectively. T 2.2 (Fontaine [14], Cherbonnier-Colmez [6], Berger [1], [2]) D(T ) (resp. D(V ), D† (V ), Drig (V )) is an étale (ϕ, Γ)-module of rank d. Furthermore, the functor T 7→ D(T ) (resp. V 7→ D(V ), V 7→ D† (V ), V 7→ Drig (V )) is a rank preserving equivalence of categories from the category of OL -linear (resp. L-linear) GQp -representations to the category of étale (ϕ, Γ)-modules over OE (resp. E , E † , R). 4 e SÉRIE – TOME 45 – 2012 – No 1.

(8) 173. LOCALLY ANALYTIC VECTORS. Let D be a (ϕ, Γ)-module over R. If D is isomorphic to its ϕ-pullback, then for any y ∈ D, Pp−1 i we may write y = i=0 (1 + T ) ϕ(yi ) for some uniquely determined yi ∈ D. We define ψ : D → D by setting ψ(y) = y0 . It follows that ψ commutes with Γ and satisfies ψ(aϕ(x)) = ψ(a)x,. ψ(ϕ(a)x) = aψ(x). for any a ∈ R, x ∈ D. In particular, ψ is a left inverse to ϕ. Set RespZp (y) = ϕψ(y), ResZ× (y) = (1−ϕψ)(y), and denote RespZp (D), ResZ× (D) by DpZp , DZ× p respectively. p p For an étale (ϕ, Γ)-module D over OE , A treillis of D is a compact OE + -submodule N which OE -linearly generates D. Colmez [11] proves that the set of ψ-stable treillis admits a unique minimal element D\ ,and that ψ is surjective on D\ . It follows from the uniqueness that D\ is stable under the Γ-action. In the simplest case when D = OE , we have D\ = OE + . For an étale (ϕ, Γ)-module D over E , if D is the base change of an étale (ϕ, Γ)-module D0 over OE , we define D\ = D0\ [1/p] which is independent of the choice of D0 . dT We define the ϕ, Γ-actions on the rank 1 R-module R 1+T formally by ϕ(x. dT dT ) = ϕ(x) , 1+T 1+T. γ(x. dT dT ) = χ(γ)γ(x) , x ∈ R. 1+T 1+T. dT Then the rank 1 (ϕ, Γ)-module R 1+T is isomorphic to R(). For any étale (ϕ, Γ)-module D dT over R, the étale (ϕ, Γ)-module Ď = HomR (D, R 1+T ) is called the Tate dual of D. We define the pairing {·, ·} : Ď × D → L by setting {x, y} = res0 (σ−1 (x)(y)). It follows that {·, ·} is perfect and satisfies. {x, ϕ(y)} = {ψ(x), y}.. (2.3). 3. p-adic local Langlands correspondence for GL2 (Qp ) 3.1. Operator wδ ×. For an étale (ϕ, Γ)-module D over OE or E and a continuous character δ : Q× p → OL , × defined by → D Z Colmez constructs the involution wδ : D Z× p p X −1 −1 i δ(i )(1 + T ) σ−i2 · ϕn ψ n ((1 + T )−i z). wδ (z) = lim (3.1) n→+∞ × i∈Zp mod pn. ×. . Since δ(Z× Note that the right hand side of (3.1) only involves δ|Z× p ) ⊆ OL for any p c δ ∈ T (L), (3.1) is still convergent for any D which is a twist of an étale (ϕ, Γ) one and c(L). From now on, we suppose D is a twist of an étale (ϕ, Γ)-module over OE or δ ∈ T c(L), and define wδ : D Z× → D Z× by (3.1). Let D† , Drig denote the E and δ ∈ T p. (ϕ, Γ)-modules over E † , R corresponding to D.. p. × E 3.1. – It follows from (3.1) that OE + Z× p ⊂ OE Zp is stable under wδ . ψ=0 more explicitly. Namely, for any By [11, V], one can describe the wδ -action on ( OE + ) + f ∈ C 0 (Zp , L) and z ∈ OE Z× , p Z Z f (x)A −1 (wδ (z)) = δ(x)f (1/x)A −1 (z). (3.2) Z× p. Z× p. ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE.

(9) 174. R. LIU, B. XIE AND Y. ZHANG. For any abelian profinite group C, we denote by ΛC the complete group algebra. OL [[C]] = ←− lim OL [C/C 0 ] where C 0 goes through all open subgroups of C. If C is pro-p cyclic, and if c is a topological generator of C, then ΛC is canonically isomorphic to the ring consisting of g(c − 1) for all g(T ) ∈ OL [[T ]], and we further define R(C) to be the ring consisting of g(c − 1) for all † g(T ) ∈ R for any R of E + , E † , OE , E , OE , R + and R; this is independent of the choice of c. Now we choose d ≥ 1 such that Γd is pro-p cyclic (in fact, we can choose d = 1 if p is odd, and d = 2 if p = 2), then we define R(Γ) = ΛΓ ⊗ΛΓd R(Γd ) which is independent of the choice of d. The topological rings OE (Γ) or E (Γ), E † (Γ) and R(Γ) naturally act on D Z× p, × D† Z× and D Z respectively. The following Lemma follows from the proof of [12, rig p p Lemme V.2.2]. −1 L 3.2. – For any γ ∈ Γ and z ∈ D Z× (wδ (z)). p , wδ (γ(z)) = δ(χ(γ))γ. Let ιδ : R(Γ) → R(Γ) denote the involution defined by γ → δ(χ(γ))γ −1 . It is an immediate consequence of Lemma 3.2 that the action of wδ on D Z× p is OE (Γ)-semilinear with respect to ιδ , i.e. (3.3). wδ (λ(z)) = ιδ (λ)(wδ (z)),. λ ∈ OE (Γ), z ∈ D Z× p.. c(L). Let η ∈ T + P 3.3. – E + (η) Z× p is a free E (Γ)-module of rank 1. Furthermore, we have + + × R + (η) Z× p = R (Γ) ⊗E + (Γ) E (η) Zp ,. † + × E † (η) Z× p = E (Γ) ⊗E + (Γ) E (η) Zp .. As a consequence, E † (η) Z× p is stable under wδ . Proof. – It suffices to treat the case η = 1. By [12, Lemme V.1.16], E † Z× p is a free + + × E (Γ)-module of rank 1 generated by 1+T . By [18, B.2.8], R Zp is a free R (Γ)-module † ψ=0 ∩ (R + )ψ=0 is a free also generated by 1 + T . We thus deduce that E + Z× p = (E ) + E (Γ)-module generated by 1 + T . The last assertion follows from (3.3) and the fact that E + Z× p is stable under wδ . †. 3.2. Construction of the correspondence We define. ¶ © × (z1 )) D δ P1 = z = (z1 , z2 ) ∈ D × D | ResZ× (z ) = w (Res , 2 δ Z p p. and we equip D δ P1 with the subspace topology of D × D. P 3.4. – There exists a unique continuous action of GL2 (Qp ) on D δ P1 satisfying the following conditions: (i) (ii) (iii) (iv). w(z1 , z2 ) = (z2 , z1 ); a 0 if a ∈ Q× p , then [ 0 a ] (z1 , z2 ) = (δ(a)z1 , δ(a)z2 ); × a if a ∈ Zp , then [ 0 01 ] (z1 , z2 ) = (σa (z1 ), δ(a)σa−1 (z2 )); if z = (z1 , z2 ) and if z 0 = p0 10 (z), then RespZp z 0 = ϕ(z1 ) and ResZp (wz 0 ) = δ(p)ψ(z2 );. 4 e SÉRIE – TOME 45 – 2012 – No 1.

(10) LOCALLY ANALYTIC VECTORS. 175. (v) for b ∈ pZp , if z = (z1 , z2 ) and if z 0 = [ 10 1b ] z, then ResZp z 0 = [ 10 1b ] z1 and RespZp (wz 0 ) = ub (RespZp (z2 )), where î ó (1+b)−2 b(1+b)−1 ◦ w ◦ 1 1/(1+b) . ◦ w ◦ ub = δ(1 + b) 10 −1 δ δ 1 0 1 0 1 Proof. – It is easy to see that the matrices given in the proposition generate GL2 (Qp ). This implies the uniqueness. The existence follows from [12, Proposition II.1.8]. (Its proof applies to our more general situation.) We extend {·, ·} : Ď × D → L to a pairing {·, ·}P1 : (Ď δ−1 P1 ) × (D δ P1 ) → L by setting (3.4). {(z1 , z2 ), (z10 , z20 )}P1 = {z1 , z10 } + {RespZp (z2 ), RespZp (z20 )}.. P 3.5. – The pairing {·, ·}P1 : (Ď δ−1 P1 ) × (D δ P1 ) → L is perfect and GL2 (Qp )-equivariant. Proof. – This follows immediately from [12, Théorème II.3.1]. 0 0 ) for some ) or E (δD Now Let D be of rank two. Then det D is of the form OE (δD −1 0 0 1 1 c δD ∈ T (L). Let δD = δD , and we denote wδD , D δD P by wD , D P for simplicity. For z = (z1 , z2 ) ∈ D P1 , set ResZp (z1 , z2 ) = z1 . We then define n D\ P1 = {z ∈ D P1 | ResZp ( p0 01 z) ∈ D\ , ∀n ∈ N}.. T 3.6 ([12, Théorème II.3.1]). – Let D be an irreducible rank 2 étale (ϕ, Γ)-module over OE . Then the following hold: (i) The submodule D\ P1 of D P1 is stable under the action of GL2 (Qp ). (ii) The quotient GL2 (Qp )-representation Π(D) = D P1 /D\ P1 is an object of Rep OL GL2 (Qp ), and has central character δD . The continuous GL2 (Qp )-representation D\ P1 is naturally isomorphic to Π(D)∗ ⊗ δD . Hence we have the following exact sequence 0 −→ Π(D)∗ ⊗ δD −→ D P1 −→ Π(D) −→ 0. −1 Note that Ď ∼ ) because D is of rank two. Hence Π(D)∗ ⊗ δD is isomorphic to = D(δD. (Π(Ď) ⊗ δD )∗ ⊗ δD = Π(Ď)∗ . C 3.7. – If D is an irreducible étale (ϕ, Γ)-module of rank 2 over E , then D\ P1 is stable under the GL2 (Qp )-action and the quotient representation Π(D) = D P1 /D\ P1 is an admissible unitary representation of GL2 (Qp ). Moreover, D\ P1 is naturally isomorphic to the contragredient representation Π(Ď)∗ . ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE.

(11) 176. R. LIU, B. XIE AND Y. ZHANG. 3.3. Locally analytic vectors Now suppose that D is a twist of an irreducible rank 2 étale (ϕ, Γ)-module over E . The following proposition follows immediately from ([12, Lemme V.2.4]). P 3.8. – D† Z× p is stable under the action of wD . † × By [12, Théorème V.1.12(iii)], we have Drig Z× p = R(Γ) ⊗E † (Γ) (D Zp ). Then for ∆ = R(η), ∗ = δ, or ∆ = Drig , ∗ = δD , we extend the w∗ -action to ∆ Z× p by setting. w∗ (λ ⊗ z) = ι∗ (λ) ⊗ w∗ (z). For ∆ = E † (η), R + (η), R(η), ∗ = δ, or ∆ = D† , Drig , ∗ = δD , we set ∆ ∗ P1 = {(z1 , z2 ) ∈ ∆ × ∆, ResZ× (z2 ) = w∗ (ResZ× (z1 ))}, p p and we equip ∆∗ P1 with the subspace topology of ∆×∆. Henceforth we denote D† δD P1 , Drig δD P1 by D† P1 , Drig P1 for simplicity. By Proposition 3.4, it is clear that both E † (η)δ P1 and D† P1 are stable under GL2 (Qp ). Since D† is dense in Drig for any (ϕ, Γ)-module D over E , we extend the GL2 (Qp )-actions on E † (η) δ P1 and D† P1 to continuous GL2 (Qp )-actions on R(η) δ P1 and Drig P1 which satisfy the formulas listed in Proposition 3.4 by continuity. This yields the following proposition. P 3.9. – There exists a unique continuous action of GL2 (Qp ) on ∆ ∗ P1 satisfying the formulas listed in Proposition 3.4. For ∆ = R(η), ∗ = δ, or ∆ = Drig , ∗ = δD , we set the pairing ˇ ∗−1 P1 × ∆ ∗ P1 → L {·, ·}P1 : ∆ by formula (3.4). ˇ ∗−1 P1 × ∆ ∗ P1 → L is perfect and P 3.10. – The pairing {·, ·}P1 : ∆ GL2 (Qp )-equivariant. Proof. – The restriction of {·, ·}P1 on E † (η̌) δ−1 P1 × E † (η) δ P1 or Ď† P1 × D† P1 is GL2 (Qp )-equivariant by Proposition 3.5. Hence {·, ·}P1 itself is GL2 (Qp )-equivariant by the density of E † (η)δ P1 or D† P1 . The perfectness of {·, ·}P1 follows from the perfectness ˇ × ∆ and ∆ ˇ pZp × ∆ pZp . of {·, ·} on ∆ If D is étale, it follows from [11, Corollaire II.7.2] that D\ ⊂ D† ; hence we may view D P1 as a submodule of Drig P1 . Colmez shows that the inclusion Π(Ď)∗ = D\ P1 ⊂ Drig P1 extends naturally to a GL2 (Qp )-equivariant embedding (Π(Ď)an )∗ ,→ Drig P1 , \ and he further shows that the image of this embedding, which is denoted by Drig P1 , is the \ 1 1 orthogonal complement of Ď P under the pairing {·, ·}P1 : Ďrig P ×Drig P1 → L ([12, Remarque V.2.21(ii)]). The following proposition is the key ingredient for our determination of locally analytic vectors of unitary principal series. \. \ \ P 3.11 ([12, Remarque V.2.21(i)]). – Drig P1 and Ďrig P1 are orthogonal complements of each other under the pairing {·, ·}P1 : Ďrig P1 × Drig P1 → L.. 4 e SÉRIE – TOME 45 – 2012 – No 1.

(12) LOCALLY ANALYTIC VECTORS. 177. 4. Unitary principal series and 2-dimensional trianguline representations 4.1. 2-dimensional irreducible trianguline representations A (ϕ, Γ)-module over R is called triangulable if it is a successive extensions of rank 1 (ϕ, Γ)-modules over R; an L-linear representation V of GQp is called trianguline if Drig (V ) is triangulable. P 4.1 ([7, Proposition 3.1]). – If D is a rank 1 (ϕ, Γ)-module over R, then c(L) such that D is isomorphic to R(δ). there exists a unique δ ∈ T It follows that if V is a 2-dimensional irreducible trianguline representation, then Drig (V ) sits in a short exact sequence (4.1). 0 −→ R(δ1 ) −→ Drig (V ) −→ R(δ2 ) −→ 0. c(L). Furthermore, we have that (4.1) is non-split by Kedlaya’s slope for some δ1 , δ2 ∈ T theory. Therefore V is uniquely determined by the triple (δ1 , δ2 , c) where c ∈ Proj(Ext1 (R(δ2 ), R(δ1 ))) = Proj(H 1 (R(δ1 δ2−1 ))) is the element representing the extension (4.1). Let c(L), c ∈ Proj(H 1 (R(δ1 δ −1 )))} S = {(δ1 , δ2 , c)|δi ∈ T 2 be the set of all such triples; then each element of S naturally defines a rank 2 triangulable (ϕ, Γ)-module: the non-split extension of R(δ2 ) by R(δ1 ) defined by c. In the rest of this section we assume p > 2. The following calculation is due to Colmez. P 4.2 ([7, Théorème 2.9]). – dimL H 1 (R(δ)) = 2. (ii) Otherwise, dimL (R(δ)) = 1.. (i) If δ = x−k or xk+1 |x| for k ∈ N, then. Colmez further specifies an explicit basis of H 1 (R(δ)) in case (i), and identifies Proj(H 1 (R(δ))) with P1 (L) via this basis. By this identification, we may write any s ∈ S as s = (δ1 , δ2 , L ) where L ∈ P1 (L) if δ1 δ2−1 = x−k or xk+1 |x| for some k ∈ N, or L = ∞ otherwise. Let D(s) denote the rank 2 triangulable (ϕ, Γ)-module defined by s. We set š ∈ S to be the triple (δˇ1 , δˇ2 , L ) = (δ2−1 , δ1−1 , L ). Then Ď(s) is isomorphic to D(š) under Colmez’s identification. Let S∗ be the set of all s = (δ1 , δ2 , L ) ∈ S such that vp (δ1 (p)) + vp (δ2 (p)) = 0,. vp (δ1 (p)) > 0.. For s ∈ S∗ , set u(s) = vp (δ1 (p)) = −vp (δ2 (p)), w(s) = w(δ1 ) − w(δ2 ), δs = δ1 δ2−1 (x|x|)−1 , and we define S∗ng = {s ∈ S∗ | w(s) is not a positive integer}, S∗cris = {s ∈ S∗ | w(s) is a positive integer, u(s) < w(s), L = ∞}, and Sirr k ∈ Z+ .. S∗st = {s ∈ S∗ | w(s) is a positive integer, u(s) < w(s), L 6= ∞}, ` ` = S∗ng S∗cris S∗st . Note that if s ∈ S∗st , we must have δs = xk−1 for some. ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE.

(13) 178. R. LIU, B. XIE AND Y. ZHANG. P 4.3 ([7, Proposition 3.5, 3.7]). – If s ∈ Sirr , then D(s) is étale, and the corresponding L-linear representation, which we denote by V (s), is irreducible. Conversely, every 2-dimensional irreducible trianguline representation is isomorphic to V (s) for some s ∈ Sirr . Moreover, for s = (δ1 , δ2 , L ), s0 = (δ10 , δ20 , L 0 ) ∈ Sirr , if δ1 = δ10 , then V (s) ∼ = V (s0 ) if and 0 0 0 0 ∼ only if δ2 = δ2 and L = L ; if δ1 6= δ1 , then V (s) = V (s ) if and only if s, s0 ∈ S∗cris and δ10 = xw(s) δ2 , δ20 = x−w(s) δ1 . We call s ∈ Sirr exceptional if s ∈ S∗cris and V (s) is not Frobenius semi-simple; this is equivalent to δs = xk−1 |x|−1 for some k ∈ Z+ . 4.2. Unitary principal series Throughout this subsection, let s = (δ1 , δ2 , L ) ∈ Sirr . Let C u (P1 (δ)) be the L-vector space of C u functions f : Qp → L such that δ(x)f (1/x)|Qp −{0} extends to a C u -function on Qp . In other words, C u(s) (P1 (δs )) is the L-vector space of functions f : Qp → L satisfying: • f |Zp is of class C u(s) . • δs (z)f (1/z)|Zp −{0} extends to a C u(s) -function on Zp . We thus have an isomorphism C u(s) (P1 (δs )) ' C u(s) (Zp , L) ⊕ C u(s) (Zp , L),. f 7→ (f1 , f2 ). where f1 (z) = f (pz) and f2 is the extension of δs (z)f (1/z). By this isomorphism, we may equip C u(s) (P1 (δs )) with a Banach space structure by defining ||f || = max ||f1 ||C u(s) , ||f2 ||C u(s) . We define a GL2 (Qp )-representation B(s) on C u(s) (P1 (δs )) by ã Å a b dx − b ; ? f (x) = δ (ad − bc)δ (−cx + a)f s 2 s c d −cx + a then B(s) is a Banach space representation. We define a subspace M (s) of B(s) as below: – If δs 6= xk for any k ∈ N, we define M (s) to be the subspace generated by {xi |0 ≤ i < u(s)} and {(x − a)−i δs (x − a)|a ∈ Qp , 0 ≤ i < u(s)}. – If δs = xk−1 for some k ∈ Z+ , let M (s)0 be the space of functions of the form X f= λu (x − au )ju logL (x − au ) u∈U. where U is a finite set, ju are integers between [ k+1 2 ] and k, λu ∈ L and au ∈ Qp P ju such that deg( u∈U λu (x − au ) ) < u(s). By [5, Lemme 3.3.2], M (s)0 is a subspace of B(s). We define M (s) to be the subspace generated by M (s)0 and xi for 0 ≤ i ≤ k − 1. An easy computation shows that M (s) is stable under GL2 (Qp ) in both cases. We set c(s) where M c(s) is the closure of M (s) in B(s). Π(s) = B(s)/M Let D\ (s) denote (D(V (s)))\ for simplicity. We fix a standard basis e2 of R(δ2 ). For any n (n) z ∈ D\ (s) P1 , suppose that the image of ResZp ( p0 01 z) in R(δ2 ) is z2 e2 . The following theorem follows from [12, Théorème IV.4.12]. 4 e SÉRIE – TOME 45 – 2012 – No 1.

(14) 179. LOCALLY ANALYTIC VECTORS. T 4.4. – For s ∈ Sirr non-exceptional and z ∈ D\ (s) P1 , there exists µz ∈ Du(s) (Qp ) such that (n). A (n) (µz ) = (δ2 (p))−n z2 . Furthermore, the map z 7→ µz is a GL2 (Qp )-equivariant topological isomorphism from D\ (s) P1 to Π(š)∗ . We denote the converse of this isomorphism by As .. 5. Locally analytic principal series and rank 1 (ϕ, Γ)-modules 5.1. Locally analytic principal series c(L), we denote by LA(P1 (δ)) the L-vector space of locally analytic For any δ ∈ T functions f : Qp → L such that δ(x)f (1/x)|Qp −{0} extends to a locally analytic function on Qp . As in the case of C u (P1 (δ)), for any f ∈ LA(P1 (δ)), if we set f1 (pz) = f |pZp and f2 to be the extension of δs (x)f (1/x)|Zp −{0} , then the map f 7→ f1 ⊕f2 induces an isomorphism LA(P1 (δ)) ∼ = LA(Zp , L) ⊕ LA(Zp , L). We then equip LA(P1 (δ)) with the topology induced from LA(Zp , L) ⊕ LA(Zp , L). c(L) × T c(L), let Σ(δ e 1 , δ2 ) denote the locally analytic parabolic For any pair (δ1 , δ2 ) ∈ T induction Ä GL (Q ) ä IndB(Q2p ) p δ2 ⊗ δ1 −1. an. = {locally analytic functions F : GL2 (Qp ) → L such that F (bg) = (δ2 ⊗ δ1 −1 )(b)F (g) for all b ∈ B(Qp )},. which is equipped with the left GL2 (Qp )-action (gF )(g 0 ) = F (g 0 g) for any g, g 0 ∈ GL2 (Qp ). e 1 , δ2 ) with Put δ = δ1 δ2−1 −1 . We may identify the underlying topological space of Σ(δ 1 LA(P (δ)) by the map 0 1 F 7→ f (x) := F ( −1 x ) e 1 , δ2 ). In addition, the corresponding GL2 (Qp )-action on LA(P1 (δ)) is given for any F ∈ Σ(δ by the formula (5.1). Å a b c d. · f (x) = δ2 (ad − bc)δ(−cx + a)f. ã dx − b . −cx + a. If k = w(δ1 δ2−1 ) = w(δ) + 1 is a positive integer, then the k-th differential map Å ãk d Ik : LA(P1 (δ)) → LA(P1 (x−2k δ)), f (x) 7→ f (x), dx e 1 , δ2 ) and Σ(x e −k δ1 , xk δ2 ). The kernel of Ik , which induces an intertwining between Σ(δ consists of locally polynomial functions of degree ≤ k − 1, is isomorphic to (5.2). GL (Q ). (δ2 ◦ det) ⊗ Symk−1 L2 ⊗ IndB(Q2p ) p (1 ⊗ (x−k+1 δ))sm. as a locally analytic representation. Moreover, if δ = xk−1 , the L-vector subspace generated by {xi |0 ≤ i ≤ k − 1} is GL2 (Qp )-invariant, and is isomorphic to (δ2 ◦ det) ⊗ Symk−1 L2 as a GL2 (Qp )-representation. The quotient of ker Ik by this subspace is isomorphic to (δ2 ◦ det) ⊗ Symk−1 L2 ⊗ St. ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE.

(15) 180. R. LIU, B. XIE AND Y. ZHANG. We define ( (5.3) Σ(δ1 , δ2 ) =. e 1 , δ2 )/(δ2 ◦ det) ⊗ Symk−1 L2 if δ = xk−1 for some integer k ≥ 1; Σ(δ e 1 , δ2 ) Σ(δ otherwise.. The following proposition, which follows by the main results of [21], [20], determines the Jordan-Hölder series of Σ(δ1 , δ2 ). P 5.1. – With notations as above, the following are true. e 1 , δ2 ) is a topological irreducible locally analytic (i) If w(δ) ∈ / N, then Σ(δ1 , δ2 ) = Σ(δ representation of GL2 (Qp ). e 1 , δ2 ) is a non-split (ii) If w(δ) ∈ N and δ 6= xk−1 , then Ik is surjective, and Σ(δ1 , δ2 ) = Σ(δ GL (Q ) 2 k−1 −k k 2 e extension of Σ(x δ1 , x δ2 ) by (δ2 ◦ det) ⊗ Sym L ⊗ IndB(Qp ) p (1 ⊗ (x−k+1 δ))sm , e −k δ1 , xk δ2 ) and (δ2 ◦ det) ⊗ Symk−1 L2 ⊗ IndGL2 (Qp ) (1 ⊗ (x−k+1 δ))sm and both Σ(x B(Qp ). are topological irreducible. (iii) If δ = xk−1 for some integer k ≥ 1, then Σ(δ1 , δ2 ) is a non-split extension e −k δ1 , xk δ2 ) by (δ2 ◦ det) ⊗ Symk−1 L2 ⊗ St, and both Σ(x e −k δ1 , xk δ2 ) and of Σ(x k−1 2 (δ2 ◦ det) ⊗ Sym L ⊗ St are topological irreducible. 5.2. Σ̃(η −1 , δ −1 η)∗ ∼ = R + (η) δ P1 . c(L), let GL2 (Qp ) acts on Σ(δ e 1 , δ2 )∗ by the formula hf, g · µi = For any δ1 , δ2 ∈ T ∗ e e hg · f, µi for any f ∈ Σ(δ1 , δ2 ), µ ∈ Σ(δ1 , δ2 ) and g ∈ GL2 (Qp ). Thus by (5.1), we have −1. (wf )(x) = η(−1)δη −2 (x)f (1/x) e −1 , δ −1 η). Therefore, by the description of LA(P1 (δ)) given in §5.1, we see for any f ∈ Σ(η e −1 , δ −1 η)∗ to that the map µ 7→ (µ|Zp , wµ|Zp ) is a homeomorphism from Σ(η Z Z (5.4) {(µ1 , µ2 ) ∈ D(Zp , L) ⊕ D(Zp , L)| f (x)µ2 = η(−1)(δη −2 )(x)f (1/x)µ1 }, Z× p. Z× p. where the latter object is equipped with the subspace topology of D(Zp , L) ⊕ D(Zp , L). We fix a standard basis eη ∈ E + (η). e −1 , δ −1 η)∗ . L 5.2. – A (wµ|Z× ) ⊗ eη = wδ (A (µ|Z× ) ⊗ eη ) for any µ ∈ Σ(η p p Proof. – The case A (µ|Z× ) ∈ E + Z× p follows directly from (3.2) and (5.4). Since p + + E is dense in R , we deduce the case A (µ|Z× ) ∈ R + Z× p by the continuity of w and p A . We then conclude the general case by Proposition 3.3. As a consequence, Aδ,η (µ) = (A (µ|Zp )⊗eη , A (wµ|Zp )⊗eη ) is an element of R + (η)δ P1 . e −1 , δ −1 η)∗ P 5.3. – The map Aδ,η : Σ(η GL2 (Qp )-equivariant topological isomorphism. 4 e SÉRIE – TOME 45 – 2012 – No 1. →. R + (η) δ P1 is a.

(16) 181. LOCALLY ANALYTIC VECTORS. e −1 , δ −1 η)∗ given in (5.4), one sees easily that Aη,δ is Proof. – By the description of Σ(η an embedding. On the other hand, for any z = (z1 ⊗ eη , z2 ⊗ eη ) ∈ R + (η) δ P1 , if we put µ = A −1 (z1 ) + wA −1 (RespZp (µ2 )), then Aη,δ (µ) = z. Hence Aη,δ is a topological isomorphism. To prove that Aη,δ is GL2 (Qp )-equivariant, we only need to show Aη,δ (g · µ) = g · Aη,δ (µ). (5.5). p 0 × a0 for (1) g = [ 01 10 ]; (2) g = [ a0 a0 ] , a ∈ Q× p ; (3) g = [ 0 1 ] , a ∈ Zp ; (4) g = 0 1 ; (5) g = [ 10 1b ] , b ∈ pZp . e −1 , δ −1 η)∗ and R + (η) δ P1 have central characters δ; this Case (1) is trivial. Both Σ(η proves (2). For any a ∈ Z× p , we have Z Z Z −1 a 0 a 0 ( 0 1 f (x))µ = f (x)([ 0 1 ])µ) =. η(a. Z. this yields (3). For case (4), we have Z Z f (x)( p0 10 µ) = pZp. Zp. î. pZp. f (x)(η(a−1 )(σa (µ)));. )f (ax)µ =. Zp. Zp. Zp. −1. p−1 0 0 1. ó. (f (x)1Zp (x))µ. Z =. η(p)f (px)µ Zp. Z =. f (x)(η(p)ϕ(µ|Zp )), Zp. yielding. p 0 0 1. (5.6). µ|pZp = η(p)ϕ(µ|Zp ). This implies RespZp (Aδ,η ( p0 10 µ)) = ϕ(ResZp (Aδ,η (µ))).. A similar computation shows that î −1 ó ResZp (w(Aδ,η ( p0 10 µ))) = ResZp (Aδ,η ( 10 p0 wµ)) = δ(p)ResZp (Aδ,η ( p 0 10 wµ)) (5.7) = δ(p)ψ(ResZp ((Aδ,η (wµ)))) = δ(p)ψ(ResZp (w(Aδ,η (µ)))). This proves (4). For case (5), first note that Z Z Z 0 1 −b0 f (x) 10 b1 µ = f (x)µ = f (x + b0 )µ. 0 1 This implies 0 0 A ( 10 b1 µ|U ) = 10 b1 A (µ|U ). (5.8). for any b0 ∈ U ⊆ Zp . Hence ResZp (Aδ,η ([ 10 1b ] µ)) = [ 10 1b ] ResZp (Aδ,η (µ)). It remains to check that (5.9). RespZp (Aδ,η (w [ 10 1b ] µ)) = ub (RespZp (Aδ,η (wµ))),. where ub = δ(1 + b). 1 −1 0 1. ◦ wδ ◦. î. (1+b)−2 b(1+b)−1 0 1. ó. ◦ wδ ◦. ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE. 1 1/(1+b) 0. 1. ..

(17) 182. R. LIU, B. XIE AND Y. ZHANG. By (5.8), Lemma 5.2 and case (2), ub (RespZp (Aδ,η (wµ))) = RespZp (Aδ,η (δ(1 + b). 1 −1 î (1+b)−2 0 1 w 0. b(1+b)−1 1. ó î 1/(1+b) 0 ó w 10 1/(1+b) w µ)) 0 1/(1+b) 1. = RespZp (Aδ,η ([ 01 1b ] µ)) = RespZp (Aδ,η (w [ 10 1b ] µ)). This proves (5.12). 6. Determination of locally analytic vectors 6.1. Σ(s) and Emerton’s conjecture We first recall the locally analytic representations Σ(k, L ) of GL2 (Qp ) which are originally constructed by Breuil (in the case L 6= ∞). We refer the reader to [4, 2.1] and [13, 5.1] for more details. Fix an integer k ≥ 2. Given L ∈ P1 (L), let σ(L ) denote the representation of B(Qp ) on L2 = Le1 ⊕ Le2 defined by a0 db e1 = e1 , a0 db e2 = e1 + (logL a − logL d)e2 . One thus has a non-split extension 0 −→ 1 −→ σ(L ) −→ 1 −→ 0.. (6.1). k−2 We put σ(k, L ) = σ(L ) ⊗ χk where χk : B(Qp ) → L× is the character a0 db 7→ |ad| 2 dk−2 . Twisting (6.1) by χk , and then taking locally analytic parabolic induction, one obtains an exact sequence of locally analytic representations GL (Q ). s. GL (Q ). GL (Q ). L (6.2) 0 −→ (IndB(Q2p ) p χk )an −→ (IndB(Q2p ) p σ(k, L ))an −→ (IndB(Q2p ) p χk )an −→ 0.. Note that χk = |x| has (|x| define (6.3). k−2 2. k−2 2. ⊗ xk−2 |x|. ◦ det) ⊗ Sym. k−2. k−2 2. GL (Q ). k. e k−1 |x| 2 , |x| . Thus (IndB(Q2p ) p χk )an = Σ(x. k−2 2. ) which. L2 as a subrepresentation following the discussion above. We. Σ(k, L ) = s−1 L ((|x|. k−2 2. ◦ det) ⊗ Symk−2 L2 )/(|x|. k−2 2. ◦ det) ⊗ Symk−2 L2 .. One thus has an extension of locally analytic representations (6.4). k. 0 −→ Σ(xk−1 |x| 2 , |x|. k−2 2. ) −→ Σ(k, L ) −→ (|x|. k−2 2. ◦ det) ⊗ Symk−2 L2 −→ 0.. From now on, let s = (δ1 , δ2 , L ) ∈ Sirr . We define (6.5) ( 2−k Σ(k + 1, L ) ⊗ ((δ2 |x| 2 ) ◦ det) if w(s) = k is a positive integer and δs = xk−1 ; Σ(s) = e 1 , δ2 ) Σ(δ otherwise. It follows that in the first case Σ(s) sits in the exact sequence (6.6). 0 −→ Σ(δ1 , δ2 ) −→ Σ(s) −→ (δ2 ◦ det) ⊗ Symk−1 L2 −→ 0.. Following [4, 2.2], we now give a geometric model of Σ(s) in the first case. Let LA(P1 (xk−1 , L )) be the space of locally analytic functions H on Qp with values in L such that +∞ X an (6.7) H(z) = z k−1 ( ) + P (z) logL (z), zn n=0 4 e SÉRIE – TOME 45 – 2012 – No 1.

(18) 183. LOCALLY ANALYTIC VECTORS. for |z| 0, where an ∈ L, P (z) is a polynomial of degree ≤ k − 1 with coefficients in L. Let GL2 (Qp ) act on this space by ï Å ãò . ([ ac db ]?s H)(z) = δ2 (ad−bc)(−cz+a)k−1 H. dz − b −cz + a. −. 1 P 2. dz − b logL −cz + a. ad − bc (−cz + a)2. .. Note that LA(P1 (xk−1 )) is exactly the subspace consisting of functions H with P = 0 in the expression (6.7). An easy computation shows that the L-vector subspace generated by xi , 0 ≤ i ≤ k − 1, is GL2 (Qp )-invariant. We define C(xk−1 , L ) to be the quotient of LA(P1 (xk−1 , L )) by this subspace. It turns out that the resulting representation of GL2 (Qp ) on C(xk−1 , L ) is topologically isomorphic to Σ(s), and the natural map LA(P1 (xk−1 )) → C(xk−1 , L ) gives rise to the inclusion Σ(δ1 , δ2 ) ,→ Σ(s). We denote by C(xk−1 ) the image of the map LA(P1 (xk−1 )) → C(xk−1 , L ). Then the quotient C(xk−1 , L )/C(xk−1 ) is a k-dimensional L-vector space spanned by 1D(∞,1) · xn logL x, 0 ≤ n ≤ k − 1, which is isomorphic to (δ2 ◦ det) ⊗ Symk−1 L2 . By this geometric model, one can show that (cf. [4, Lemme 2.4.2]) (δ2 ◦det)⊗Symk−1 L2 ⊗St (resp. (δ2 ◦det)⊗Symk−1 L2 ) is the only topologically irreducible subrepresentation (resp. quotient representation) of Σ(s). In particular, the extension (6.6) is non-split. Although it is known to experts that there is a natural morphism Σ(s) → Π(s) which realizes Π(s) as the universal completion of Σ(s), we cannot find a reference for this result. For our purpose, we rephrase the work of Breuil and Emerton in the following proposition to construct the desired morphism. We first note that the natural inclusion LA(P1 (δs )) ⊂ C u(s) (P1 (δs )) induces a GL2 (Qp )-equivariant continuous map e 1 , δ2 ) → Π(s), ιs : Σ(δ. f 7→ f¯.. P 6.1. – For s ∈ Sirr non-exceptional, the GL2 (Qp )-equivariant continuous e 1 , δ2 ) → Π(s) induces an injection ιs : Σ(δ1 , δ2 ) → Π(s). Moreover, in the case map ιs : Σ(δ when δs = xk−1 for some positive integer k, the map ιs : Σ(δ1 , δ2 ) → Π(s) extends naturally to an injective map ιs : Σ(s) → Π(s) which is continuous and GL2 (Qp )-equivariant. Proof. – If δs = xk−1 , since the subrepresentation (δ2 ◦ det) ⊗ Symk−1 L2 , which consists of polynomials of degree ≤ k − 1, is contained in M (s), ιs induces a map Σ(δ1 , δ2 ) → Π(s). The injectivity of ιs on Σ(δ1 , δ2 ) is proved by Emerton in [13, Lemma 6.7.2]. We rewrite his proof in our set up as below for the reader’s convenience. We first have that ιs (Σ(δ1 , δ2 )) is dense in Π(s) since LA(P1 (δs )) is dense in C u(s) (P1 (δs )). Hence ιs is nonzero because Π(s) 6= 0 by Theorem 4.4. If w(δs ) ∈ / N, Σ(δ1 , δ2 ) is topologically irreducible by Proposition 5.1. Thus ιs is either injective or zero. It therefore follows that ιs must be injective. In case w(δs ) ∈ N, we put k = w(s) = w(δs ) + 1. We see from Proposition 5.1 that all the proper admissible subrepresentations of Σ(δ1 , δ2 ) are contained in the image of ker(Ik ). Thus it reduces to show that ιs is injective on the image of ker(Ik ). Note that k = w(s) > u(s). Therefore LP[0,k−1] (Zp , L), the space of locally polynomial functions of degree ≤ k − 1 on Zp , is dense in C u(s) (Zp , L) by the classical theorem of Amice-Vélu and Vishik. We thus deduce that ιs (ker(Ik )) is dense in Π(s). It follows that ιs (ker(Ik )) is infinite dimensional because Π(s) is infinite dimensional. If δs 6= xk−1 , the only possible nontrivial quotient ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE.

(19) 184. R. LIU, B. XIE AND Y. ZHANG. of ker(Ik ) is the finite dimensional representation (δ2 |x|−1 ◦det)⊗Symk−1 L2 ). Hence ιs must be injective on ker(Ik ) in this case. If δs = xk−1 , note that the image of ker(Ik ) in Σ(δ1 , δ2 ), which is isomorphic to (δ2 ◦ det) ⊗ Symk−1 L2 , is irreducible. It follows that ιs is injective on the image of ker(Ik ) as well. Now suppose δs = xk−1 . The extension of ιs to Σ(s) is actually due to Breuil who identifies Π(s) with the universal unitary completion of Σ(s) and shows that the natural map Σ(s) → Π(s) is injective as long as Π(s) 6= 0 ([4, Proposition 4.3.5], [5, Corollaire 3.3.4]). We briefly recall his construction of the natural map Σ(s) → Π(s) as below. One easily sees that Breuil’s map extends ιs . For any 0 ≤ i ≤ k − 1, and X li (x) = λu (x − au )ju logL (x − au ) u∈U. where U is a finite set, ju are integers between [ k+1 2 ] and k − 1, λu ∈ L, au ∈ Qp P such that deg( u∈U λu (x − au )ju + xi ) < u(s), it follows from [5, Lemme 3.3.2] that li (x) + xi logL (x)1D(∞,n) ∈ C u(s) (P1 (δs )) for n ∈ Z. We thus define Z Z (li (x) + xi logL (x)1D(∞,n) )µ(x) xi logL (x)µ(x) = P1 (Qp ). D(∞,n). for any µ ∈ Π(s)∗ ; this is independent of the choice of li (x) because the difference of any such two li0 s lies in M (s)0 which is killed by µ. By this way, we extend µ to an element of C(δs , L )∗ . This yields a continuous GL2 (Qp )-equivariant morphism Π(s)∗ → Σ(s)∗ . Taking dual of this morphism, we get Breuil’s map Σ(s) → Π(s). We are now in the position to reformulate Emerton’s conjecture for non-exceptional s. Note that ιs (Σ(s)) ⊂ Π(s)an since Σ(s) is a locally analytic representation. C 6.2. – For s ∈ Sirr non-exceptional, the cokernel of the inclusion ιs : e 2 , δ1 ) as locally analytic GL2 (Qp )-representations. Thus Σ(s) → Π(s)an is isomorphic to Σ(δ the space of locally analytic vectors Π(s)an sits in a short exact sequence of locally analytic GL2 (Qp )-representations e 2 , δ1 ) −→ 0. 0 −→ Σ(s) −→ Π(s)an −→ Σ(δ. (6.8). R 6.3. – Emerton shows that if the above conjecture is true, then the extension (6.8) must be non-split ([13]). In the case when s ∈ S∗cris , there is a more explicit description of Π(s)an which is due to Breuil. Recall that D(s) is isomorphic to D(s0 ) for s0 = (xw(s) δ2 , x−w(s) δ1 , L ). We thus obtain a morphism Σ((xw(s) δ2 , x−w(s) δ1 ) → Π(s0 )an ∼ = Π(s)an . On the other hand, if × × α, β : Qp → L are smooth characters such that |α(p)| ≤ |β(p)|, there is an intertwining GL (Q ). GL (Q ). from IndB(Q2p ) p (α ⊗ β|x|−1 )sm to IndB(Q2p ) p (β ⊗ α|x|−1 )sm , yielding an intertwining from GL (Q ). GL (Q ). Symk−1 L2 ⊗ IndB(Q2p ) p (α ⊗ β|x|−1 )sm to Symk−1 L2 ⊗ IndB(Q2p ) p (β ⊗ α|x|−1 )sm . It thus follows that if we set Σ(δ1 , δ2 )lalg to be the image of ker Ik in Σ(δ1 , δ2 ), then there exists an intertwining between Σ(δ1 , δ2 )lalg and Σ(xw(s) δ2 , x−w(s) δ1 )lalg which is always injective (but the direction can be either way). We therefore get a morphism (6.9). Σ(δ1 , δ2 ) ‹ ⊕ Σ(xw(s) δ2 , x−w(s) δ1 ) → Π(s)an. 4 e SÉRIE – TOME 45 – 2012 – No 1.

(20) LOCALLY ANALYTIC VECTORS. 185. where ‹ ⊕ denotes the amalgamated sum of two summands over the intertwining between Σ(δ1 , δ2 )lalg and Σ(xw(s) δ2 , x−w(s) δ1 )lalg . C 6.4 ([3, Conjectures 4.4.1, 5.3.7]). – For s ∈ S∗cris non-exceptional, (6.9) is a topological isomorphism. P 6.5. – For s ∈ S∗cris non-exceptional, Emerton’s conjecture is equivalent to Breuil’s conjecture. Proof. – The generic case that V (s) does not admit an L -invariant (this is equivalent to δs 6= xw(s)−1 , xw(s)−1 |x|−2 ) is already proved in [13, 6.7.5]. We now prove the remaining cases. The injectivity of (6.9) is already ensured by [3, Corollaires 5.3.6, e 2 , δ1 ) have 5.4.3]. It reduces to show that Σ(δ1 , δ2 ) ‹ ⊕ Σ(xw(s) δ2 , x−w(s) δ1 ) and Σ(s) ⊕ Σ(δ w(s)−1 −2 same constitutes. If δs = x |x| , then Σ(s) = Σ(δ1 , δ2 ) and the intertwining is Σ(xw(s) δ2 , x−w(s) δ1 )lalg → Σ(δ1 , δ2 )lalg . Therefore Ä ä Σ(δ1 , δ2 ) ‹ ⊕ Σ(xw(s) δ2 , x−w(s) δ1 ) /Σ(s) e 2 , δ1 ) = Σ(xw(s) δ2 , x−w(s) δ1 )/Σ(xw(s) δ2 , x−w(s) δ1 )lalg ∼ = Σ(δ by Proposition 5.1. If δs = xw(s)−1 , the intertwining is Σ(δ1 , δ2 )lalg → Σ(xw(s) δ2 , x−w(s) δ1 )lalg and the quotient is isomorphic to (δ2 ◦ det) ⊗ Symw(s)−1 L2 . Thus Ä ä Σ(δ1 , δ2 ) ‹ ⊕ Σ(xw(s) δ2 , x−w(s) δ1 ) /Σ(δ1 , δ2 ) ∼ e 2 , δ1 ) by is an extension of Σ(xw(s) δ2 , x−w(s) δ1 )/Σ(xw(s) δ2 , x−w(s) δ1 )lalg Σ(δ = w(s)−1 2 w(s) −w(s) ‹ (δ2 ⊗ det) ⊗ Sym L . We thus obtain that Σ(δ1 , δ2 ) ⊕ Σ(x δ2 , x δ1 ) and e 2 , δ1 ) have same constitutes in both cases. Σ(s) ⊕ Σ(δ 6.2. An exact sequence From now on, we suppose p > 2. Recall that for any s = (δ1 , δ2 , L ) ∈ Sirr , there is an exact sequence i. j. 0 −→ R(δ1 ) −→ D(s) −→ R(δ2 ) −→ 0. For i = 1, 2, we denote Aδs ,δi , R + (δi ) δs P1 , R(δi ) δs P1 by Ai , R + (δi ) P1 , R(δi ) P1 for simplicity. Recall that As : Π(š)∗ → D\ (s) P1 is the topological GL2 (Qp )-equivariant isomorphism given by Theorem 4.4. P 6.6. – If s is non-exceptional, then the GL2 (Qp )-equivariant morphism A2 ◦ ι∗š ◦ As−1 : D\ (s) P1 → R + (δ2 ) P1 satisfies A2 ◦ ι∗š ◦ As−1 ((z, z 0 )) = (j(z), j(z 0 )) for any (z, z 0 ) ∈ D\ (s) P1 . Proof. – Suppose that the images of z, z 0 in R(δ2 ) are z2 e2 , z20 e2 where e2 is the standard basis of R(δ2 ) fixed in §4.2. Suppose As−1 ((z, z 0 )) = µ. Then by the definition of As , we see that A (µ|Zp ) = z2 , A (wµ|Zp ) = z20 . Hence A2 ◦ ι∗s ◦ As−1 ((z, z 0 )) = (z2 ⊗ e2 , z20 ⊗ e2 ) = (j(z), j(z 0 )). C 6.7. – If s is non-exceptional, then j(wD (x)) = wδs (j(x)) for any x ∈ D(s) Z× p. ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE.

(21) 186. R. LIU, B. XIE AND Y. ZHANG. Proof. – If x ∈ (1 − ϕ)D(s)ψ=1 , by the proof of [12, Proposition V.2.1], there exists z = (z1 , z2 ) ∈ D\ (s) P1 such that ResZ× z1 = x. Since j commutes with ResZ× and p p + 1 (j(z1 ), j(z2 )) ∈ R (δ2 ) P , we get wδs (j(x)) = wδs (j(ResZ× (z1 ))) = wδs (ResZ× (j(z1 ))) p p = ResZ× (j(z2 )) = j(ResZ× (z2 )) = j(wD (x)). p p ψ=1 By [12, Corollaire V.1.13], D(s) Z× as an R(Γ)-module. p is generated by (1 − ϕ)D(s) ψ=1 We conclude the corollary from the case x ∈ (1 − ϕ)D(s) and the fact that both wD , wδs are R(Γ)-antilinear.. P 6.8. – The maps iP1 : R(δ1 ) P1 → D(s) P1 , (z1 , z2 ) 7→ (i(z1 ), i(z2 )) and jP1 : D(s) P1 → R(δ2 ) P1 , (z1 , z2 ) 7→ (j(z1 ), j(z2 )) are well-defined morphisms of continuous GL2 (Qp )-representations. Moreover, we have the short exact sequence (6.10). i. 1. j. 1. P P 0 −→ R(δ1 ) P1 −→ D(s) P1 −→ R(δ2 ) P1 −→ 0.. Proof. – By Propositions 3.4, 3.9, the GL2 (Qp )-actions on R(δ1 ) P1 , R(δ2 ) P1 and D(s) P1 satisfy the same set of formulas. It thus follows that iP1 , jP1 are GL2 (Qp )-equivariant as long as they are well-defined. In general, the formula (3.1) of wD is not convergent on D† Z× p . Hence the well-defineness of iP1 and jP1 are not obvious from their definitions. The well-defineness of jP1 follows from Corollary 6.7. To show that iP1 is well-defined, we use the pairing {·, ·}. First note that i : R(δ1 ) → D(s) is dual to j : D(š) → R(δ̌1 ) × with respect to h·, ·i. It therefore follows that i : R(δ1 ) Z× p → D(s) Zp is the dual × × × of j : D(š) Z× p → R(δ̌1 ) Zp with respect to h·, ·i. Hence i : R(δ1 ) Zp → D(s) Zp × × is the dual of j : D(š) Zp → R(δ̌1 ) Zp with respect to {·, ·}. Since {·, ·} is w-invariant and we have already proved that j commutes with w on D(š) Z× p , we thus deduce that × 1 i commutes with w on R(δ1 ) Zp . Hence iP1 : R(δ1 ) P → D(s) P1 is well-defined. For (6.10), the injectivity of iP1 and the exactness at D(s) P1 are obvious. To show the surjectivity of jP1 , for any (z, z 0 ) ∈ R(δ2 ) P1 , we pick y ∈ D(s) and y 0 ∈ D(s) pZp which lift z and RespZp z 0 respectively. Then an easy computation shows that y)) = (z, z 0 ). jP1 (y, y 0 + wD (ResZ× p This proves that jP1 is surjective. Henceforth we identify R(δ1 ) P1 with a submodule of D(s) P1 via iP1 . 4 e SÉRIE – TOME 45 – 2012 – No 1.

(22) LOCALLY ANALYTIC VECTORS. 187. 6.3. Proof of Emerton’s conjecture We prove Conjecture 6.2 for p > 2 in this subsection. From now on, let s ∈ Sirr be non-exceptional. Since As is a topological GL2 (Qp )-equivariant isomorphism between the contragredient representation Π(š)∗ and D\ (š) P1 , it induces an isomorphism \ As,an : (Π(š)an )∗ → ((D\ (s) P1 )an )∗ = Drig (s) P1 \ between coadmissible D(GL2 (Zp ))-modules (Π(š)an )∗ and Drig (s)P1 , where D(GL2 (Zp )) denotes the algebra of locally analytic distributions on GL2 (Zp ).. P 6.9. – The diagram (Π(š)an )∗. (6.11). As,an. / D\ (s) P1 rig. ι∗ š. e δˇ2 , δˇ1 )∗ Σ(. jP1. A2. / R(δ2 ) P1. \ is commutative. As a consequence, we have jP1 (Drig (s) P1 ) = A2 (Σ(δˇ2 , δˇ1 )∗ ).. Proof. – Recall that for an admissible Banach space representation U of GL2 (Qp ), Uan ∗ . The diagram (6.11) commutes is dense in U ([22, Theorem 7.1]); hence U ∗ is dense in Uan ∗ ∗ on Π(š) ⊂ (Π(š)an ) following Proposition 6.6. We thus conclude the commutativity \ of (6.11) by the density of Π(š)∗ in (Π(š)an )∗ . It thus follows that jP1 (Drig (s) P1 ) = ∗ ∗ ∗ A2 (ιš ((Π(š)an ) )) = A2 (Σ(δˇ2 , δˇ1 ) ). L 6.10. – R + (η̌) δ−1 P1 and R + (η) δ P1 are orthogonal complements of each other under the pairing R(η̌) δ−1 P1 × R(η) δ P1 → L. Proof. – It suffices to show that R + is the orthogonal complements of itself under the pairing {·, ·} : R × R → L. It is obvious that {R + , R + } = 0. On the other hand, if P f = i∈Z ai T i ∈ (R + )⊥ , then for any j ∈ N, {σ−1 (T j ), f } = a−j−1 implies a−j−1 = 0, yielding f ∈ R + . \ \ L 6.11. – jP1 (Drig (s) P1 ) ⊂ R(δ2 ) P1 and Drig (š) P1 ∩ R(δ̌2 ) P1 are orthogonal complements of each other under {·, ·}P1 : R(δ2 ) P1 × R(δ̌2 ) P1 → L.. Proof. – By the constructions of iP1 and jP1 , one easily checks that iP1 : R(δˇ2 ) P1 → D(š) P1 is dual to jP1 : D(s) P1 → R(δ2 ) P1 with respect to {·, ·}P1 . Thus by Proposition 3.11, we deduce that {jP1 (x), y}P1 = {x, y}P1 = 0 \ and y ∈ Drig (š) P1 ∩ R(δˇ2 ) P1 . This proves P1 ∩ R(δˇ2 ) P1 )⊥ . e δˇ1 , δˇ2 ) are admissible locally analytic repOn the other hand, since Σ(δˇ1 , δˇ2 ) and Σ( ∗ ∗ ˇ ˇ ˇ ˇ e resentations, Σ(δ1 , δ2 ) and Σ(δ1 , δ2 ) are coadmissible D(GL2 (Zp ))-modules. Theree δˇ1 , δˇ2 )∗ by [22, Lemma 3.6]. This implies that fore Σ(δˇ1 , δˇ2 )∗ is a closed subspace of Σ( \ jP1 (D (š) P1 ) = A2 (Σ(δˇ1 , δˇ2 )∗ ) is a closed subspace of R + (δ̌1 ) P1 by Proposition 6.9;. for any x \ jP1 (Drig (s). \ (s) P1 ∈ Drig \ P1 ) ⊆ (Drig (š) . rig. \ (š) P1 ) is Fréchet complete with the subspace topology of R(δ̌1 ) P1 . By hence jP1 (Drig. ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE.

(23) 188. R. LIU, B. XIE AND Y. ZHANG. the open mapping theorem for Fréchet type spaces ([19, Proposition 8.8]), we deduce that \ \ jP1 : Drig (š) P1 → jP1 (Drig (š) P1 ) is open. Therefore the quotient topology and the \ subspace topology on jP1 (Drig (š) P1 ) coincide. Now for any x ∈ (D\ (š) P1 ∩ R(δˇ2 ) P1 )⊥ ⊂ R(δ2 ) P1 , we pick x̃ ∈ D(s) P1 rig. \ (š) P1 induces a lifting x. The continuous linear functional f (y) = {x̃, y}P1 on Drig \ 1 continuous linear functional f¯ on jP1 (Drig (š) P ). Applying Hahn-Banach theorem for Fréchet type spaces ([19, Corollary 9.4]), we extend f¯ to a continuous linear functional on R(δˇ1 ) P1 . Since the pairing R(δˇ1 ) P1 × R(δ1 ) P1 → L is perfect, we may suppose that the extension of f¯ is defined by some x0 ∈ R(δ1 ) P1 . It therefore follows that for any \ y ∈ Drig (š) P1 ,. {x̃ − x0 , y}P1 = {x̃, y}P1 − {x0 , jP1 (y)}P1 = f¯(jP1 (y)) − {x0 , jP1 (y)}P1 = 0, \ \ (s) P1 ) because yielding x̃ − x0 ∈ Drig (s) P1 . We thus conclude that x ∈ jP1 (Drig \ \ 0 1 1 ⊥ ˇ jP1 (x̃ − x ) = x. This proves (Drig (š) P ∩ R(δ2 ) P ) ⊆ jP1 (Drig (s) P1 ).. P 6.12. – The following are true: \ (i) if δs = xk−1 , where k is an integer ≥ 2, then R(δ1 ) P1 ∩ Drig (s) P1 contains \ R + (δ1 ) P1 as a closed subspace of codimension k, and jP1 (Drig (s) P1 ) is a closed + 1 subspace of R (δ2 ) P of codimension k; \ \ (ii) otherwise, R(δ1 ) P1 ∩ Drig (s) P1 = R + (δ1 ) P1 and jP1 (Drig (s) P1 ) = + 1 R (δ2 ) P .. Proof. – We prove (i) only. The proof of (ii) is similar. For (i), it follows from Propo\ sition 6.9 that jP1 (Drig (s) P1 ) = A2 (Σ(δˇ2 , δˇ1 )∗ ). Recall that Σ(δˇ2 , δˇ1 ) is a quotient e δˇ2 , δˇ1 ) by a k-dimensional subrepresentation. Hence Σ(δˇ2 , δˇ1 )∗ is a closed subof Σ( e δˇ2 , δˇ1 )∗ of codimension k, yielding that jP1 (D\ (s) P1 ) is a closed subspace space of Σ( rig. of R + (δ2 ) P1 of codimension k. On the other hand, as \ \ R(δ1 ) P1 ∩ Drig (s) P1 = jP1 (Drig (š) P1 )⊥. and. R + (δ1 ) P1 = (R + (δ̌1 ) P1 )⊥. by Lemmas 6.11, 6.10, we deduce that R + (δ1 ) P1 is a codimension k closed subspace \ of R(δ1 ) P1 ∩ Drig (s). T 6.13. – Conjecture 6.2 is true for p > 2. \ (š) P1 . Let Σ be the Proof. – By Proposition 6.12, R + (δˇ2 ) P1 is contained in Drig \ ∗ locally analytic representation such that Σ is isomorphic to Drig (š) P1 /R + (δˇ2 ) δ̌ P1 . e 2 , δ1 )∗ , we thus have an exact sequence of locally Since R + (δˇ2 ) P1 is isomorphic to Σ(δ analytic L-representations of GL2 (Qp ). (6.12). ι1 ι2 e 0 −→ Σ −→ Π(s)an −→ Σ(δ2 , δ1 ) −→ 0.. If δs is not of the form xk−1 for any k ∈ Z+ , then Σ∗ ∼ = R + (δˇ1 ) P1 by Proposition 6.12(ii), e which in turn is isomorphic to the dual of Σ(δ1 , δ2 ). We thus have that Σ is isomorphic e 1 , δ2 ) = Σ(s), yielding (6.8) in this case. Now suppose δs = xk−1 for some integer to Σ(δ e 2 , δ1 ) is topologically irreducible, and it is not isomorphic to any topological k ≥ 1. Since Σ(δ 4 e SÉRIE – TOME 45 – 2012 – No 1.